29.29.24 problem 846
Internal
problem
ID
[5429]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
29
Problem
number
:
846
Date
solved
:
Sunday, March 30, 2025 at 08:13:00 AM
CAS
classification
:
[[_homogeneous, `class A`], _rational, _dAlembert]
\begin{align*} x {y^{\prime }}^{2}+x -2 y&=0 \end{align*}
✓ Maple. Time used: 0.028 (sec). Leaf size: 96
ode:=x*diff(y(x),x)^2+x-2*y(x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {\left (2 \operatorname {LambertW}\left (\frac {\sqrt {c_1 x}}{c_1}\right )^{2}+2 \operatorname {LambertW}\left (\frac {\sqrt {c_1 x}}{c_1}\right )+1\right ) x}{2 \operatorname {LambertW}\left (\frac {\sqrt {c_1 x}}{c_1}\right )^{2}} \\
y &= \frac {\left (2 \operatorname {LambertW}\left (-\frac {\sqrt {c_1 x}}{c_1}\right )^{2}+2 \operatorname {LambertW}\left (-\frac {\sqrt {c_1 x}}{c_1}\right )+1\right ) x}{2 \operatorname {LambertW}\left (-\frac {\sqrt {c_1 x}}{c_1}\right )^{2}} \\
\end{align*}
✓ Mathematica. Time used: 0.583 (sec). Leaf size: 97
ode=x (D[y[x],x])^2+x-2 y[x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
\text {Solve}\left [\frac {2}{\sqrt {\frac {2 y(x)}{x}-1}-1}-2 \log \left (\sqrt {\frac {2 y(x)}{x}-1}-1\right )&=\log (x)+c_1,y(x)\right ] \\
\text {Solve}\left [\frac {2}{\sqrt {\frac {2 y(x)}{x}-1}+1}+2 \log \left (\sqrt {\frac {2 y(x)}{x}-1}+1\right )&=-\log (x)+c_1,y(x)\right ] \\
\end{align*}
✓ Sympy. Time used: 6.925 (sec). Leaf size: 75
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x*Derivative(y(x), x)**2 + x - 2*y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ \log {\left (x \right )} = C_{1} - \log {\left (- \sqrt {-1 + \frac {2 y{\left (x \right )}}{x}} + \frac {y{\left (x \right )}}{x} \right )} + \frac {2}{\sqrt {-1 + \frac {2 y{\left (x \right )}}{x}} - 1}, \ \log {\left (x \right )} = C_{1} - \log {\left (\sqrt {-1 + \frac {2 y{\left (x \right )}}{x}} + \frac {y{\left (x \right )}}{x} \right )} - \frac {2}{\sqrt {-1 + \frac {2 y{\left (x \right )}}{x}} + 1}\right ]
\]